Question: $\dfrac{ -3e + f }{ -8 } = \dfrac{ -e + 5g }{ -5 }$ Solve for $e$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -3e + f }{ -{8} } = \dfrac{ -e + 5g }{ -5 }$ $-{8} \cdot \dfrac{ -3e + f }{ -{8} } = -{8} \cdot \dfrac{ -e + 5g }{ -5 }$ $-3e + f = -{8} \cdot \dfrac { -e + 5g }{ -5 }$ Multiply both sides by the right denominator. $-3e + f = -8 \cdot \dfrac{ -e + 5g }{ -{5} }$ $-{5} \cdot \left( -3e + f \right) = -{5} \cdot -8 \cdot \dfrac{ -e + 5g }{ -{5} }$ $-{5} \cdot \left( -3e + f \right) = -8 \cdot \left( -e + 5g \right)$ Distribute both sides $-{5} \cdot \left( -3e + f \right) = -{8} \cdot \left( -e + 5g \right)$ ${15}e - {5}f = {8}e - {40}g$ Combine $e$ terms on the left. ${15e} - 5f = {8e} - 40g$ ${7e} - 5f = -40g$ Move the $f$ term to the right. $7e - {5f} = -40g$ $7e = -40g + {5f}$ Isolate $e$ by dividing both sides by its coefficient. ${7}e = -40g + 5f$ $e = \dfrac{ -40g + 5f }{ {7} }$